Definition 3.5.6 In a rotating body, a vector Ω is called an angular velocity vector if the velocity of a point having position vector u relative to the body is given by Ω × u.
The existence of an angular velocity vector is the key to understanding motion in a moving system of coordinates. It is used to explain the motion on the surface of the rotating earth. For example, have you ever wondered why low pressure areas rotate counter clockwise in the upper hemisphere but clockwise in the lower hemisphere? To quantify these things, you will need the concept of an angular velocity vector. Details are presented later for interesting examples. Here is a simple example. In the above example, think of a coordinate system fixed in the rotating body. Thus if you were riding on the rotating body, you would observe this coordinate system as fixed but it is not fixed.
Example 3.5.7 A wheel rotates counter clockwise about the vector i + j + k at 60 revolutions per minute. This means that if the thumb of your right hand were to point in the direction of i + j + k your fingers of this hand would wrap in the direction of rotation. Find the angular velocity vector for this wheel. Assume the unit of distance is meters and the unit of time is minutes.
Let ω = 60 × 2π = 120π. This is the number of radians per minute corresponding to 60 revolutions per minute. Then the angular velocity vector is
Example 3.5.8 A wheel rotates counter clockwise about the vector i + j + k at 60 revolutions per minute exactly as in Example 3.5.7. Let {u_{1},u_{2},u_{3}} denote an orthogonal right handed system attached to the rotating wheel in which u_{3} =
Since {u_{1},u_{2},u_{3}} is a right handed system like i,j,k, everything applies to this system in the same way as with i,j,k. Thus the cross product is given by

Therefore, in terms of the given vectors u_{i}, the angular velocity vector is 120πu_{3}. The velocity of the given point is

in meters per minute. Note how this gives the answer in terms of these vectors which are fixed in the body, not in space. Since u_{i} depends on t, this shows the answer in this case does also. Of course this is right. Just think of what is going on with the wheel rotating. Those vectors which are fixed in the wheel are moving in space. The velocity of a point in the wheel should be constantly changing. However, its speed will not change. The speed will be the magnitude of the velocity and this is

which from the properties of the dot product equals

because the u_{i} are given to be orthogonal.